In this paper, we establish the following result: Let $(T,{ \cal F},\mu)$ be a $\sigma$-finite measure space, let $Y$ be a reflexive real Banach space, and let $\varphi, \psi:Y\to { \bf R}$ be two sequentially weakly lower semicontinuous functionals such that $$\inf_{ y\in Y}{ { \min\{ \varphi(y),\psi(y)\}}\over { 1+\|y\|^p}}>-\infty$$ for some $p>0$. Moreover, assume that $\varphi$ has no global minima, while $\varphi+\lambda\psi$ is coercive and has a unique global minimum for each $\lambda>0$. Then, for each $\gamma\in L^{ \infty}(T)\cap L^1(T)\setminus \{ 0\}$, with $\gamma\geq 0$, and for each $r>\inf_{ Y}\psi$, if we put $$V_{ \gamma,r}= \left \{ u\in L^p(T,Y) : \int_T\gamma(t)\psi(u(t))d\mu\leq r\int_T\gamma(t)d\mu\right \}\ ,$$ we have $$\inf_{ u\in V_{ \gamma,r}} \int_T\gamma(t)\varphi(u(t))d\mu= \inf_{ \psi^{ -1}(r)}\varphi\int_T\gamma(t)d\mu\ .$$
Categories: math.OC